Complex coordinate transformations and the Schwarzschild--Kerr metrics

Utilizing the fact that the Schwarzschild and Kerr geometries have an intrinsically defined Minkowski space associated with them, we show that these Minkowski spaces (as the Kerr parameter varies) can be viewed as ``real slices'' in a complexified Minkowski space. The complex Weyl tensor of each member of the family can then be viewed as a single complex field on the complex Minkowski space. Further, the degenerate principle null vectors associated with each geometry can be considered as projections into the ``real slices'' of a complex null vector field in the complex Minkowski space. These results may be considered as clarifying earlier work on obtaining the Kerr metric from the Schwarzschild metric by a complex coordinate transformation.